Integrating Shell Collapse Velocity Equation for Free-Fall Time Calculation

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The discussion centers on deriving the free-fall time of a collapsing star using the shell collapse velocity equation. The equation provided is \(\frac{dr}{dt}=-[GM(\frac{1}{2}-\frac{1}{R})]^{1/2}\), and the user seeks to integrate it by substituting \(x=r/R\) to utilize a standard integral. There is confusion regarding the right-hand side of the equation, with concerns that it may not be correctly expressed as a function of \(r\). The conversation highlights the challenges in manipulating the equation for integration. Overall, the focus is on finding the correct approach to solve the integral for free-fall time.
TheTourist
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I have derived the following equation for the velocity of a collapsing shell of a collapsing star.
\frac{dr}{dt}=-[GM(\frac{1}{2}-\frac{1}{R})]1/2​

I now need to integrate this to find the free-fall time, and the hint is to manipulate the equation into a form where you can use a standard integral: substitute x=r/R and then use the standard integral \int[\frac{x}{1-x}]1/2 dx=\pi/2

I am completely stuck on how to do this
 
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Hi TheTourist! :smile:

(have an integral: ∫ and a square-root: √ and a pi: π :wink:)

hmm … that RHS can't possibly be correct :redface: …

shouldn't it be a function of r? :smile:​
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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